Reprints in Theory and Applications of Categories, No. 8, 2005, pp. 1–24.
TAKING CATEGORIES SERIOUSLY
F. WILLIAM LAWVERE
Abstract. The relation between teaching and research is partly embodied in simple general concepts which can guide the elaboration of examples in both. Notions and con- structions, such as the spectral analysis of dynamical systems, have important aspects that can be understood and pursued without the complication of limiting the models to specific classical categories. Pursuing that idea leads to a dynamical objectification of Dedekind reals which is particularly suited to the simple identification of metric spaces as enriched categories over a special closed category. Rejecting the complacent descrip- tion of that identification as a mere analogy or amusement, its relentless pursuit [8] is continued, revealing convexity and geodesics as concepts having a definite meaning over any closed category. Along the way various hopefully enlightening exercises for students (and possible directions for research) are inevitably encountered: (1) an explicit treat- ment of the contrast between multiplication and divisibility that, in inexorable functorial fashion, mutates into the adjoint relation between autonomous and non-autonomous dy- namical systems; (2) the role of commutation relations in the contrast between equilibria and orbits, as well as in qualitative distinctions between extensions of Heyting logic; (3) the functorial contrast between translations and rotations (as appropriately defined) in an arbitrary non symmetric metric space.
The theory of categories originated [1] with the need to guide complicated calculations involving passage to the limit in the study of the qualitative leap from spaces to homo- topical/homological objects. Since then it is still actively used for those problems but also in algebraic geometry [2], logic and set theory [3], model theory [4], functional analysis [5], continuum physics [6], combinatorics [7], etc. In all these the categorical concept of adjoint functor has come to play a key role. Such a universal instrument for guiding the learning, development, and use of advanced mathematics does not fail to have its indications also in areas of school and college mathe- matics, in the most basic relationships of space and quantity and the calculations based on those relationships. In saying “take categories seriously”, I advocate noticing, cultivating,
After the original manuscript was stolen, I was able to reconstruct, in time for inclusion in the proceedings of the 1983 Bogot´a Workshop, this“sketch” (as Saunders Mac Lane’s review was to describe it). Interested teachers can elaborate this and similar material into texts for beginning students. For the present opportunity, I am grateful to Xavier Caicedo who gave permission to reprint, to Christoph Schubert for his expert transcription into TEX, and to the editors of TAC. Received by the editors 2004-11-02. Transmitted by M. Barr, R. Rosebrugh and R. F. C. Walters. Reprint published on 2005-03-23. 2000 Mathematics Subject Classification: 06D20, 18D23, 37B55, 34L05, 52A99, 53C22, 54E25, 54E40. Key words and phrases: General spectral theory, Nonautonomous dynamical systems, Symmetric monoidal categories, Semimetric spaces, General convexity, Geodesics, Heyting algebras, Special maps on metric spaces. The article originally appeared in Revista Colombiana de Matem´aticas, XX (1986) 147-178. c Re- vista Colombiana de Matem´aticas, 1986, used by permission. 1 2 F. WILLIAM LAWVERE and teaching of helpful examples of an elementary nature.
1. Elementary mutability of dynamical systems Already in [1] it was pointed out that a preordered set is just a category with at most one morphism between any given pair of objects, and that functors between two such cate- gories are just order-preserving maps; at the opposite extreme, a monoid is just a category with exactly one object, and functors between two such categories are just homomorphisms of monoids. But category theory does not rest content with mere classification in the spirit of Wolffian metaphysics (although a few of its practitioners may do so); rather it is the mutability of mathematically precise structures (by morphisms) which is the essential content of category theory. If the structures are themselves categories, this mutability is expressed by functors, while if the structures are functors, the mutability is expressed by natural transformations. Thus if Λ is a preordered set and X is any category (for exam- ple the category of sets and mappings, the category of topological spaces and continuous mappings, the category of linear spaces and linear transformations, or the category of bornological linear spaces and bounded linear transformations) then there are functors
Λ −→ X sometimes called “direct systems” in X , and the natural transformations
' Λ 7X
between two such functors are the appropriate morphisms for the study of such direct systems as objects. An important special case is that where Λ = 0 → 1 , the ordinal number 2, then the functors 2 −→ X may be identified with the morphisms in the category X itself; likewise if Λ = 0 → 1 → 2 →··· is the ordinal number ω, functors
X ω −−→X
are just sequences of objects and morphisms
X0 −→ X1 −→ X2 −→ X3 −→ ...
f fn in X , and a natural transformation X −−→ Y between two such is a sequence Xn −−→ Yn of morphisms in X for which all squares
fn / Xn Yn
/ Xn+1 Yn+1 fn+1 TAKING CATEGORIES SERIOUSLY 3
commute in X (here the vertical maps are the ones given as part of the structure of X and Y ).
Similarly, if M is a monoid then the functors M −→ X are extremely important math- ematical objects often known as actions of M on objects of X (or representations of M by X -endomorphisms,or...) andthenaturaltransformationsbetweensuchactionsare known variously as M-equivariant maps, intertwining operators, homogeneous functions, etc. depending on the traditions of various contexts. Historically the notion of monoid (or of group in particular) was abstracted from the actions, a pivotally important abstraction since as soon as a particular action is constructed or noticed, the demands of learning, development, and use mutate it into: 1) other actions on the same object, 2) actions on X other related objects, and 3) actions of related monoids. For if M −−→X is an action and h M −−→ M is a homomorphism, then (composition of functors!) Xh is an action of M , C while if X −−→Yis a functor, then CX is an action of M on objects of Y. To exemplify, if M is the additive group of time-translations, then a functor M −→ X is often called a dynamical system (continuous-time and autonomous) in X , but if we are interested in h observing the system only on a daily basis we could consider a homomorphism M −−→ M where M = N is the additive monoid of natural numbers, and concentrate attention on the predictions of the discrete-time, autonomous, future-directed dynamical system Xh. In other applications we might have M = M = the multiplicative monoid of real num- ()p bers, but consider the homomorphism M −−−→ M of raising to the pth power; then if we / f are given two actions M /X on objects of X , a natural transformation X −−→ (Y )p is just a morphism of the underlying X -objects which satisfies
f(λx)=λpf(x)
x for all λ in M and all T −−→ X in X , i.e. a function homogeneous of degree p.An extremely important example of the second mutation of action mentioned above is that in which Y is the opposite of an appropriate category of algebras and the functor C assigns to each object (domain of variation) of X an algebra of functions (= intensive quantities) on it. Then the induced action CX of M describes the evolution of intensive quantities which results from the evolution of “states” as described by the action X. A frequently-occurring example of the third type of mutation of action arises from the surjective homomorphisms M −→ M from the additive monoid of time-translations M X to the circle group M. Then a dynamical system M −−→X is said to be “periodic of period h” if there exists a commutative diagram of functors as follows: